# Denoise First, Orthogonalize Later: Understanding Momentum in Muon via Spectral Filtering

arXiv: https://arxiv.org/abs/2606.03899 · HTML: https://mlsanity.com/paper/2606.03899/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 61 references that are in the map.
- First posted on arXiv 2 Jun 2026.

## Builds on (most cited first, 20 of 20)

- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Gradient Descent Happens in a Tiny Subspace](https://mlsanity.com/paper/1812.04754/) (2018)
- [Fantastic Pretraining Optimizers and Where to Find Them](https://mlsanity.com/paper/2509.02046/) (2025)
- [Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization](https://mlsanity.com/paper/2503.12645/) (2025)
- [Benchmarking Optimizers for Large Language Model Pretraining](https://mlsanity.com/paper/2509.01440/) (2025)
- [Muon Outperforms Adam in Tail-End Associative Memory Learning](https://mlsanity.com/paper/2509.26030/) (2025)
- [Isotropic Curvature Model for Understanding Deep Learning Optimization: Is Gradient Orthogonalization Optimal?](https://mlsanity.com/paper/2511.00674/) (2025)
- [When do spectral gradient updates help in deep learning?](https://mlsanity.com/paper/2512.04299/) (2025)
- [Preconditioning Benefits of Spectral Orthogonalization in Muon](https://mlsanity.com/paper/2601.13474/) (2026)
- [Orthogonalising gradients to speed up neural network optimisation](https://mlsanity.com/paper/2202.07052/) (2022)
- [Momentum via Primal Averaging: Theoretical Insights and Learning Rate Schedules for Non-Convex Optimization](https://mlsanity.com/paper/2010.00406/) (2020)
- [How Muon's Spectral Design Benefits Generalization: A Study on Imbalanced Data](https://mlsanity.com/paper/2510.22980/) (2025)
- [Low-rank Orthogonalization for Large-scale Matrix Optimization with Applications to Foundation Model Training](https://mlsanity.com/paper/2509.11983/) (2025)
- [RMNP: Row-Momentum Normalized Preconditioning for Scalable Matrix-Based Optimization](https://mlsanity.com/paper/2603.20527/) (2026)
- [Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory](https://mlsanity.com/paper/2603.26554/) (2026)
- [Muon in Associative Memory Learning: Training Dynamics and Scaling Laws](https://mlsanity.com/paper/2602.05725/) (2026)
- [Spectral Gradient Descent Mitigates Anisotropy-Driven Misalignment: A Case Study in Phase Retrieval](https://mlsanity.com/paper/2601.22652/) (2026)
- [Pion: A Spectrum-Preserving Optimizer via Orthogonal Equivalence Transformation](https://mlsanity.com/paper/2605.12492/) (2026)

## Built on by (most cited first, 2 of 2)

- [Muon with Finite Newton-Schulz: The Smoothing Benefit in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2608.26288/) (2026)
- [Muon Under Gradient Noise and the Limits of Orthogonalization Near Optima](https://mlsanity.com/paper/2609.32861/) (2026)

## Abstract (arXiv)

Muon has recently demonstrated strong empirical performance in large language model training, but the theoretical role of momentum in Muon remains unclear. Existing analyses of Muon either remove momentum to study spectral updates in isolation, or retain momentum without explaining why it improves empirical performance. Our work bridges this gap by showing momentum in Muon acts as a spectral filter. Under a structured signal-plus-perturbation gradient model, we prove that momentum suppresses perturbations while preserving the dominant signal, thereby enlarging the spectral gap between them. This enlarged gap stabilizes the singular subspaces of the matrix passed to Muon's orthogonalization step, making the resulting update more reliable. We further show that applying momentum before orthogonalization achieves provably stronger alignment with the signal component of the gradient than either reversing this order or simply removing momentum. Experiments across diverse tasks, including LLM pretraining, support our theoretical analysis. More broadly, our theory offers a starting point for understanding the benefits of momentum in other matrix-based optimizers.